L(s) = 1 | + 3-s − 3·7-s + 9-s + 5·11-s − 13-s + 5·17-s + 4·19-s − 3·21-s − 2·23-s + 27-s − 9·29-s + 3·31-s + 5·33-s + 10·37-s − 39-s − 12·41-s − 2·43-s − 9·47-s + 2·49-s + 5·51-s − 9·53-s + 4·57-s + 3·59-s − 7·61-s − 3·63-s − 9·67-s − 2·69-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.13·7-s + 1/3·9-s + 1.50·11-s − 0.277·13-s + 1.21·17-s + 0.917·19-s − 0.654·21-s − 0.417·23-s + 0.192·27-s − 1.67·29-s + 0.538·31-s + 0.870·33-s + 1.64·37-s − 0.160·39-s − 1.87·41-s − 0.304·43-s − 1.31·47-s + 2/7·49-s + 0.700·51-s − 1.23·53-s + 0.529·57-s + 0.390·59-s − 0.896·61-s − 0.377·63-s − 1.09·67-s − 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.22249936687952, −14.80070942549435, −14.28256728630336, −13.87994049858174, −13.17893736499907, −12.84266938475838, −12.14665442539698, −11.70543934418052, −11.27686124876051, −10.24146191280443, −9.817017057713700, −9.478940189565012, −9.081601278371063, −8.238992927871516, −7.714230300475693, −7.147049974307984, −6.478777654316654, −6.099455680461284, −5.331172631154903, −4.551942492247865, −3.735290119210190, −3.388000353833976, −2.851460143436357, −1.754989649706163, −1.183036604268919, 0,
1.183036604268919, 1.754989649706163, 2.851460143436357, 3.388000353833976, 3.735290119210190, 4.551942492247865, 5.331172631154903, 6.099455680461284, 6.478777654316654, 7.147049974307984, 7.714230300475693, 8.238992927871516, 9.081601278371063, 9.478940189565012, 9.817017057713700, 10.24146191280443, 11.27686124876051, 11.70543934418052, 12.14665442539698, 12.84266938475838, 13.17893736499907, 13.87994049858174, 14.28256728630336, 14.80070942549435, 15.22249936687952